AMAT 342 Lecture II 10 1 19

Today Metric Spaces Continuedmore examples

Open sets and continuity

Recall A metric space is a pair Sid whereS is a set and d Sxs QD is a function such that1 dLx y O iff y2 day duy3 dG z s du DtdCy z

Limps Freon last time

The usual Euclidean metric dz on 112

ditsy t.E.lxiyitdiiasesoe.m.etmiemfn.ec

The taxi b metric d on HR

delayT Hi yil a.k.a the l metric

Example STR dm RMR o.o

dmaxlxyt maxfxi yl.kz yd Hu yn1

This is also ametric

yd max LX y

Fact If M IT is a metric space Sc Mand as S xS CO N is the restriction ofdm to 5 5 i.e d Lxy dMxy f x yesthen S ds is a metric spaceThat is subspaces of metric spaces inherit thestructure ofa metric space in the obvious way

Note In applications the subsets are often finite

But in many cases there is another construction of a metricon a subspace the intrinsic metric

Example Define a metric DonS bydLx y minimum length of an

arc in 5 connecting x and y

This is called the intrinsic metric on S

e ga d41,0 CO1 f Iz because

minimum length of an arc from 1,0 to Q1 iscircumference of S1 YI s Iz

By comparison dz Cbo COD Ftl try lengthof thestraightline connecting11,0 and10,1

Moregenerally the intrinsic metriccan be defined ona very large class of subsets 5412 as follows

differentiable

ckxy minimum length of arpath Vi ISS fromX to Y sincecodomainof r is S int is required to lie in S

As in calculus length t Soft A Idt

For example we can take S to be a sphere in 1123

it test

or any other surface in IR

Facet On S 4122 theintrinsic metric given bythegeneraldefinition is equal to the versionforSt defined earlierThis fact is proven in more generality in a course on

differential geometry

Exampleofametricspacefranbiologi

Background A DNA molecule consists ofof two chains of subunitsthe subunits are called nucleotidesthere are four nucleotidesCytosine denoted C Adenine AGuanine G Thymine T

The two chains are bound together by weak hydrogen bonds

a

The ith nucleotides in the two chains are bondedG binds to C Abonds to T Tns one chain determines theother

Illustration C G A A T A Cschematic

G C T T A T G

can represent thisSolid lines covalent bonds strong more compactly asDashedlines hydrogen bonds weak TA ffdsefence

bottomchain is determinedbythetop

The t o chains wind around each other forming a doublehelix

Fundamentalquestian Howdo we quantify thesimilarity between two DNA sequences

This is relevant to the study of evolutionclose relatives should have similar DNAdistant relatives should have dissimilar DNA

classical solution Use the edit metric

Before giving the definition let's motivate it withexamples

EI Consider the two DNA sequences

GATT G C

cantothe if stay

spots

their distance is 2

Ed CG ATT GC

cocaineoFoenedifeferenebhtthe.info gp.eto say that the distance is I

Let S denote the set consisting of sequences ofthe letters A C G T of any length 70

For XGS an elementeyoperation on X is anyone of the following operations

replace one letter in The sequence by a different one

remove one letterfromany one position in thesequentadd one letter at any one position inthesequel

Definitionoftheeditdistance o

define deal 5 5 o.o by

fine dedit S S L J b

dedit y minimum number of elementary operationsneed to transform x into y

Lecture endedhere

LetsverifythatthisisametriciaProperty 1 is clearly satisfiedAn elementary operation can always be undoneby an elementary operation so deditltykdexi.ly xIf X is a sequence of melt opstransforming X into y andB is a sequence of h elt ops

transforming y into 2 then x followed by Bis a sequence of Mtu ett ops transforminginto 2 We can choose a B sit

let's duly and molly 2 Then a followedbydenote B is a sequence of DHy t d ly z elt opsdedit asd transforming x into z It now follows that

da z s dcx.yjtdl.kz MD

Examples tfAfA deditchy 4atmost one T can becreated perelop

x ACTG

y GACTdedit Cxyj zACTG ACTS GACT

Remade Thedefinitionofedit distance generalizes toany set of symbols For example the set ofsymbols could be the entire alphabet Then theproblem ofspell checking a string of letters x canbeformalized

very naively as the problem of fining aword in the dictionary closest in edit distanceto X

tidemark Note that deceit is integer valued

Anotherexampleofametricspacefranbiologi

Background The primary function of DNA is toserve as a blue print from which proteinsare constructed

simplified definition of e proteinA protein is a string of subunits called aminoacids connected by covalent bonds

There are 20 different amino acids with nameslike arginine lysine and tryptophan

Protein fold into complex 3 D structures 9e withessential biological function e.g enzymes neurotransmitters

l o g l f m C g i 1

DNA sequences called Genies specify the aminoacid sequence of protein

tough explanat Three nucleotides specifyone amino acid

Exi C GA TT TA CCun w

Hr

Alanine Lysine Tryptophan

Determining the amino acid sequence fromtheDNAsequence is veryeasy

accuratelyBut determining The 3 D structure of the oroteinfrom the amino acid sequence ischallenging

This is called the protein structure prediction problemone of The fundamental problems ofcomputational biologyapplications to drug discoveryannual competions on this problemlots of software available